Question

The polynomials ax³ + 3x² − 3 and 2x³ − 5x + a when divided by (x − 4) leave the remainders R₁ and R₂ respectively. Find the value of the following case, if R₁ = R_2.

Answer 1

Let us denote the given polynomials as

`f(x) = ax^3 + 3x^2 -3`

`g(x) = 2x^3 - 5x + a,`

` h(x) = x-4`

Now, we will find the remainders R₁and R₂ when f(x) and g(x)respectively are divided by h(x).

By the remainder theorem, when f(x)is divided by h(x) the remainder is

`R_1 = f(4)`

` = a(4)^3 + 3(4)^2 -3`

` = 64a + 48 - 3`

` = 64a + 48`

By the remainder theorem, when g(x) is divided by h(x) the remainder is

`R_2 = g(4)`

`2(4)^3 - 5(4) + a`

`128 - 20`

` a+108`

By the given condition,

R₁ = R₂

`⇒ 64a + 45 = a + 108`

`⇒     64a - a = 108 - 45`

`⇒                 63a = 63`

`⇒                          a = 63/63 `

`⇒                              a=1`